Triple-Girder Model for Modal Analysis of Cable-Stayed Bridges with Warping Effect L.D

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Engineering Structures 22 (2000) 1313–1323 www.elsevier.com/locate/engstruct

Triple-girder model for modal analysis of cable-stayed bridges with warping effect L.D. Zhu a, H.F. Xiang a, Y.L. Xu b,* a College of Civil Engineering, Tongji University, Shanghai 200092, People’s Republic of China b Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 9 February 1999; received in revised form 19 July 1999; accepted 29 July 1999

Abstract

In the modal analysis of cable-stayed bridges using the ﬁnite element approach, the single-girder model or the double-girder model is often adopted for modeling the bridge deck. These two models are simple and easy to embody in commercial software packages, but the warping stiffness (constant) of bridge deck cannot be properly taken into consideration. This paper thus presents a triple-girder model consisting of one central girder and two side girders symmetrically connected to each other by transverse links. The proposed triple-girder model can easily consider the warping stiffness and other section properties of the bridge deck, and at the same time keep the user-friendly features in the single-girder model. The Nanpu cable-stayed bridge, built recently in China, is then taken as a case study to verify the rationality of the proposed triple-girder model through a comparison with measured data. The study shows that the modal properties of the cable-stayed bridge obtained from the modal analysis using the triple-girder model are more reasonable and close to the measured data. Further studies of four more cable-stayed bridges in China highlight that the extent of the warping effect depends greatly on the type of bridge deck and on the type of bridge tower–cable system. A simple approach for estimating some section properties of typical open-section decks is also suggested to further facilitate the use of the triple-girder model.  2000 Elsevier Science Ltd. All rights reserved.

Keywords: Cable-stayed bridge; Modal analysis; Triple-girder model; Warping constant; Case study

1. Introduction walled element model have been developed to model the bridge deck [1–7]. Increasing attention has been paid to the dynamic In the single-girder beam element model, the bridge design of cable-stayed bridges in recent years as this type deck is represented by a single beam and the cross-sec- of bridge becomes more and more popular. The dynamic tion properties of the bridge deck are assigned to the design of cable-stayed bridges subject to wind and earth- beam as equivalent properties. Conventional beam quake loading depends largely on knowledge of the elements of 12 degrees of freedom are normally used. bridges’ modal properties. Modal analysis is therefore This model is suitable for cable-stayed bridges of rela- the first important step towards a successful dynamic tively large pure torsional stiffness so that warping stiff- design. Since modern cable-stayed bridges involve a var- ness can be neglected in the equivalent beam. For cable- iety of decks, towers and cables that are connected stayed bridges with double cable planes and an open- together in different ways, the finite element method is section deck, the pure torsional stiffness of the bridge commonly regarded as the most proper way for con- deck may be small and the warping stiffness may ducting the modal analysis. In this connection, the sin- become important for the modal properties of the bridge. gle-girder beam element model, the double-girder beam This, however, presents a difficult task for the single- element model, the shell element model and the thin- girder model. To take the warping stiffness into account in the single-girder beam element model, Wilson and Gravelle [8] introduced an equivalent pure torsional con- eq * Corresponding author. Tel.: + 852-2766-6050; fax: + 852- stant Jd such that: 2334-6389. ϭ eq⌽Ј ϭ ⌽Ј Ϫ ⌽ЈЈЈ E-mail address: [email protected] (Y.L. Xu). T GJd GJd EJw , (1)

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where T is the applied torsional moment, Jd is the pure grams to carry out the modal analysis of cable-stayed torsional constant of the transformed deck cross-section bridges that are not available in most commercial com- of one material, Jw is the warping constant of the trans- puter packages. Nevertheless, when analyzing wind- formed deck cross-section, E is the modulus of elasticity induced lateral–torsional buckling and flutter problems of the transformed material, G is the shear modulus of of long-span cable-stayed bridges, large twist defor- the transformed material, and ⌽ is the torsional mode mation of the bridge deck and thus geometric non-linear shape of the bridge deck. A prime indicates the first- analysis should be considered, so that thin-walled order derivative with respect to position coordinate. elements may have to be applied [7]. Shell elements are By assuming that torsional mode functions are sine sometimes used to model a bridge deck. In this case, functions, they derived the equivalent pure torsional con- warping effects can be properly considered [3]. How- stant as follows: ever, extremely large computational effort is required for long-span cable-stayed bridges. EJ np 2 eq ϭ ϩ wͩ ͪ To overcome the shortcomings in the various models Jd Jd , (2) G L mentioned above, this paper presents a triple-girder where n is the torsional mode number (n = 1, 2, %) and model consisting of one central girder and two side gir- L is the length of the main span of the bridge. Appar- ders connected by transverse links to include the warping ently, the equivalent pure torsional constant depends on stiffness. Conventional beam elements of 12 degrees of the torsional mode number. Wilson and Gravelle [8] cal- freedom are used to model all girders, and thus the user- culated the equivalent pure torsional constant for each friendly features in the single-girder beam element of the first three torsional modes and used the average model remain. The Nanpu cable-stayed bridge built as the final equivalent constant for the bridge deck. This recently in China is taken as a case study to verify the approximation, together with the assumption of a sine rationality of the proposed triple-girder model through a torsional mode function, result in inconvenience to the comparison with measured data. After the comparison, modal analysis and uncertainties in the modal proper- four more cable-stayed bridges are analyzed to investi- ties obtained. gate the relationships between the warping effect and In the double-girder beam element model, two equiv- the type of bridge structural system. A simple way of alent beams connected by transverse links are respect- calculating some section properties for typical open-sec- ively located in each cable plane [2]. The warping stiff- tion decks is then suggested to further facilitate use of ness of the bridge deck can be considered through the the triple-girder model. opposite vertical bending stiffness of the two girders. However, the vertical bending stiffness of the bridge deck should be taken also by the vertical bending stiff- 2. Triple-girder model ness of the two girders if the transverse links are rigid and do not provide any section properties. This treatment Let us consider a typical open cross-section of bridge then leads to uncertainties in the equivalence of warping deck in cable-stayed bridges with double cable planes stiffness and vertical bending stiffness, as well as in the [Fig. 1(a)]. The cross-section should be seen as the trans- equivalence of lateral bending stiffness and longitudinal formed section for one material obtained by well-known stiffness. If the transverse links are modeled as elastic techniques as discussed, for example, by Craig [9]. The members taking some section properties of the bridge section properties of the deck are defined as follows: A deck, the equivalence of the vertical, lateral and torsional is the cross-sectional area; A¯ y is the shear area in the y- stiffnesses will be difficult to execute, and the compu- axis; A¯ z is the shear area in the z-axis; Jy is the second tation efforts for modal analysis will be increased. Fur- moment of area about the neutral y-axis; Jz is the second thermore, because the double-girder beam element moment of area about the neutral z-axis; Jd is the pure model becomes a shear type of structure in the lateral torsional constant of the cross-section; and Jw is the direction, the properties and behavior of the original warping constant with respect to the shear center. The bridge deck will be distorted to some extent. material properties are denoted as: E is the modulus of Another approach to consider warping effects in elasticity of the transformed material; G is the shear modal analysis of cable-stayed bridges is to use thin- modulus of the transformed material. walled elements to model the bridge deck. Compared The aforementioned bridge deck is now represented with the conventional beam element, the thin-walled by a triple-girder model as shown in Figs. 1(b) and (c). element has an additional warping degree of freedom at In the triple-girder model, a central girder (No. 1) is each end, expressed by the rate of twist. The elastic and located at the centroid of the original bridge deck. Two geometric stiffness matrices with or without joint side girders (No. 2) of the same section properties are rotation of a thin-walled element can be found in the symmetrically located at the corresponding cable planes. literature [5,6]. Since each thin-walled element has 14 The three girders are connected to each other through degrees of freedom, it requires special finite element pro- rigid transverse links. The rigid transverse links, how- 中国科技论文在线 http://www.paper.edu.cn

L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 1315

the second moment of area about the z-axis and the pure

torsional constant as A1, A¯ y1, A¯ z1, Jy1, Jz1 and Jd1 for the central girder, and A2, A¯ y2, A¯ z2, Jy2, Jz2 and Jd2 for each side girder. The relationship between the applied torsional moment T and the angular displacement q for the original bridge deck [see Fig. 2(a)] is then: ϭ ЈϪ ЈЈЈ T GJdq EJwq . (3)

For the equivalent triple-girder model, the equilibrium condition [see Fig. 2(b)] gives: ϭ ϩ ϩ T 2Md2 Md1 2Q2b, (4)

where Md1 and Md2 are the pure torsional moments of the central girder and one side girder, respectively; Q2 is the shearing force in the side girder in the y-direction; and b is the distance between the central girder centroid and the side girder centroid. The warping torsional moment of the side girder itself is taken as zero in Eq. (4). The pure torsional moments of the central girder and side girders in Eq. (4) are related to the angular displacement of the cross-section as: ϭ Ј ϭ Ј Md1 GJd1q ; Md2 GJd2q . (5)

Fig. 1. Schematic diagram of triple-girder model: (a) bridge deck; (b) triple-girder model; (c) bridge with triple-girder model.

ever, do not provide restrictions to the rotation of the cross-section of the side girders about the z-axis. This rotation must be independent of that of the central girder. Correspondingly, if the nodes on the central girder are taken as master nodes in the assembly of elements, the nodes on the two side girders should be partial slave nodes to the master nodes. The degree of freedom at one node controlling the rotation of the side girder about its z-axis must be an independent degree of freedom, while the other ﬁve degrees of freedom at the node should obey the corresponding master node.

2.1. Equivalent stiffness

Let us ﬁnd the equivalent stiffness of the triple-girder model for the original bridge deck. Denote the area, the shear area for the y-direction, the shear area for the z- Fig. 2. Force diagram of bridge deck section: (a) bridge deck; (b) direction, the second moment of area about the y-axis, triple-girder model. 中国科技论文在线 http://www.paper.edu.cn

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The shearing force in the side girder can be and expressed as: ϭ ϭ A¯ z2 0, A¯ z1 A¯ z. (16) ϭϪ ٞ ϭϪ ٞ Q2 EJz2y2 EJz2bq . (6)

The substitution of Eqs. (5) and (6) into Eq. (4) 2.2. Equivalent mass leads to: A proper distribution of the mass and mass moment ϭ ϩ ЈϪ 2 ٞ T G(Jd1 2Jd2)q 2EJz2b q . (7) of inertia of the original bridge deck can lead to the following equivalent masses for the central girder and the Comparison of the above equation with Eq. (3) two side girders in the triple-girder model [see Fig. 2(b)]: results in: ϭ 2 M2 IM/(2b ) (17) ϩ ϭ Jd1 2Jd2 Jd (8) and ϭ Ϫ ϭ Ϫ 2 and M1 M 2M2 M IM/b , (18)

Jw where M and I are the mass and mass moment of iner- J ϭ . (9) M z2 2b2 tia, respectively, of the original deck of certain length; M1 is the corresponding equivalent mass concentrated on the central girder; and M2 is the mass concentrated on Consideration of the vertical bending stiffness equiv- each side girder. alence between the triple-girder model and the original bridge deck gives: J 3. Case study J ϭ J Ϫ 2J ϭ J Ϫ w. (10) z1 z z2 z b2 The Nanpu cable-stayed bridge in Shanghai over the Huangpu River is taken as a case study to verify the The pure torsional constant Jd of the original bridge rationality of the proposed triple-girder model and to deck can be relatively arbitrarily distributed among the investigate the warping effect. The triple-girder model three girders in the triple-girder model according to Eq. described above and the single-girder model without (8) if only the transverse symmetry of the two side gir- warping stiffness, respectively, are incorporated into the ders is retained. The second moments of area about the finite element model of the whole bridge. Fig. 3 shows z-axis of the three girders in the triple-girder model can the finite element model of Nanpu Bridge with the triple- be determined by the second moment of area about the girder model. Since the master–slave node relation is z-axis and the warping constant of the original bridge adopted in the modal analysis, the transverse links for deck according to Eqs. (9) and (10). The equivalence of the deck model do not appear in Fig. 3. The equivalent the axial stiffness and lateral bending stiffness of the pure torsional stiffness, vertical bending stiffness and original bridge to the triple-girder model can be satisfied warping stiffness, axial stiffness and lateral bending by using the following relations: stiffness, and shear stiffness, of the three girders in the triple-girder model are determined by using Eq. (8), Eqs. A ϩ 2A ϭ A (11) 1 2 (9) and (10), Eqs. (13) and (14), and Eqs. (15) and (16), and ϩ ϩ 2 ϭ Jy1 2Jy2 2A2b Jy. (12)

Eqs. (11) and (12) may be further simpliﬁed as: ϭ ϭ A2 0, A1 A (13) and ϭ ϭ Jy2 0, Jy1 Jy. (14)

Similarly, the equivalent of the shear area can be done by setting: ϭ ϭ A¯ y2 0, A¯ y1 A¯ y (15) Fig. 3. Finite element model of Nanpu Bridge. 中国科技论文在线 http://www.paper.edu.cn

L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 1317 respectively. The modal analysis is first carried out to Table 1 lists the first 10 natural frequencies and the determine the natural frequencies and mode shapes of nature of the associated mode shapes of the Nanpu the bridge. The modal properties computed from the sin- Bridge obtained by using the single-girder model and gle-girder model and the triple-girder model are com- the triple-girder model. It is seen that the lowest natural pared with each other and then with the measured data. frequency corresponds to the first longitudinal vibration. Finally, some comments are given with respect to the This is because the Nanpu bridge deck is suspended effects of warping stiffness on modal properties of the from the bridge towers. It is also seen that all the natural bridge. frequencies concerned corresponding to the longitudinal, Nanpu Bridge is the first cable-stayed bridge of steel vertical and lateral vibration modes, obtained by using and concrete composite deck in China, and was com- the triple-girder model, are almost the same as those pleted in 1991. The total length of the bridge is 765 m obtained by using the single-girder model. However, the whereas the main span of the bridge is 423 m. Its two first torsional frequency related to the first symmetric concrete towers are 150 m high and H-shaped, support- torsional vibration mode and the second torsional fre- ing the bridge deck through suspenders in conjunction quency matching the first anti-symmetric torsional with two parallel cable planes. The 30.35 m wide bridge vibration mode are 0.4989 Hz and 0.6103 Hz, respect- deck carries two carriageways, each having three traffic ively, by the triple-girder model but 0.4218 Hz and lanes. It consists of two steel main I-section girders 0.4443 Hz, respectively, by the single-girder model. As 24.44 m apart, a series of transverse steel beams con- a result, by considering the warping stiffness in the tri- necting the two girders, and pre-fabricated concrete slabs ple-girder model, the first torsional frequency is mounted on the I-section girders and transverse beams increased by 18.3% and the second torsional frequency (see Fig. 4). is increased by 37.4%. Furthermore, it is found that with- Conventional linear elastic beam elements with 12 out the warping stiffness in the single-girder model, degrees of freedom each are used to model the girders some local torsional frequencies and mode shapes occur of Nanpu Bridge after the equivalent properties of either around the midspan of the deck and near the bridge tow- the single-girder model or the triple-girder model are ers where no stayed cables exist. However, with the determined. The conventional beam elements are also warping stiffness included in the triple-girder model, adopted to represent the bridge towers and piers based there are no such local torsional frequencies and the on the gross cross-section properties. The cables are associated mode shapes. modeled as linear elastic truss elements having an equiv- The first and second vertical vibration modes, the first alent modulus of elasticity to consider the non-linear lateral vibration mode and the first and second torsional behavior of cables caused by cable tension and sag. The modes of the Nanpu Bridge using the triple-girder model equivalent modulus of elasticity is determined using the and the single-girder model are presented in Fig. 5. The parameters of dead load cable stress, cable mass, cable solid lines and dashed lines in Fig. 5 represent the mode length and true modulus of elasticity. The suspenders shapes obtained with the triple-girder model and the sin- connecting the towers to the bridge deck are also mod- gle-girder model, respectively. It is seen that the first and eled by cable elements so as to allow free longitudinal second vertical mode shapes and the first lateral mode motion of the deck relative to towers. The lateral motion shapes from the single-girder model are almost the same of the deck relative to the towers is restricted through as those from the triple-girder model. The first and horizontal rigid links. At the abutments, the longitudinal, second torsional mode shapes from the single-girder vertical and lateral movements of the bridge deck and model are, however, different from those of the triple- the rotation about the x-axis are restrained whilst the girder model. Apparently, by including the warping con- rotations about the z-axis and the y-axis are allowed. The stant, the torsional mode shapes are much smoother in longitudinal constraints at the abutments are designed to the regions where no stayed cables exist. consider temperature effects and seismic effects. The ambient vibration measurements were carried out

Fig. 4. Typical cross-section of the deck of Nanpu Bridge. 中国科技论文在线 http://www.paper.edu.cn

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Table 1 Computed natural frequencies of Nanpu Bridge

Single-girder model Triple-girder model

Mode no. Frequency (Hz) Nature of mode shapea Mode no. Frequency (Hz) Nature of mode shapea

1 0.1427 BD ASY LG 1 0.1426 BD ASY LG 2 0.3429 BD SY LT 2 0.3431 BD SY VT 3 0.3431 BD SY VT 3 0.3460 BD SY LT 4 0.4218 BD SY TR 4 0.4235 BD ASY VT 5 0.4235 BD ASY VT 5 0.4989 BD SY TR 6 0.4443 BD ASY TR 6 0.5069 BT ASY LT BT SY LT + BD SY 7 0.5069 BT ASY LT 7 0.5391 TR BT SY LT + BD SY 8 0.5330 8 0.6103 BD ASY TR TR 9 0.5661 NDT TR 9 0.6336 BD H SY VT 10 0.5726 NDM TR 10 0.7334 BD H ASY VT

a BD — bridge deck; BT — bridge tower; LG — longitudinal; VT — vertical; LT — lateral; TR — torsional; SY — symmetric; ASY — anti- symmetric. NDM — non-cable zone at deck midspan; NDT — non-cable zone near bridge tower; H — higher.

by the Department of Bridge Engineering of Tongji Uni- versity to determine the first few natural frequencies of the completed Nanpu Bridge. Details of the arrangement of the field measurement, the instrumentation, the environment during the measurement, the signal pro- cessing and the data analysis can be found in the literature [10]. Table 2 shows a comparison of the five natural frequencies between the measured data and the computed results from the triple-girder model. These five natural frequencies correspond to the first two vertical vibration modes, the first lateral vibration mode, and the first two torsional vibration modes. Clearly, the computed torsional frequencies using the triple-girder model are much more close to the measured results than those by using the single-girder model. There is also a good agreement between the natural frequencies related to the vertical and lateral vibration modes for the computed and measured results. No local torsional frequencies and mode shapes, which are observed in the single-girder model, were found from measurements in the frequency range concerned. In summary, the good agreement between the computed and measured results for the Nanpu Bridge verifies the rationality and accuracy of the triple-girder model to some extent. Since the Nanpu Bridge has two parallel cable planes and the bridge deck is an open section with two I girders, the pure torsional stiffness of the bridge is fairly small. The warping stiffness should therefore be included in the modal analysis.

4. Parametric study and simplified calculation Fig. 5. Five important mode shapes of the Nanpu Bridge: (a) first symmetric vertical mode; (b) first symmetric lateral mode; (c) first anti-symmetric vertical mode; (d) first symmetric torsional mode; (e) In the triple-girder model, the second moment of area first anti-symmetric torsional mode. about the z-axis Jz2 of the two side girders is determined 中国科技论文在线 http://www.paper.edu.cn

L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 1319

Table 2 Comparison of the natural frequencies of the Nanpu Bridge

Mode shape Computed frequency (Hz) Measured frequency (Hz) Deviation (%)

First vertical symmetric mode 0.3431 0.360 Ϫ 4.7 First lateral symmetric mode 0.3460 0.370 Ϫ 6.5 First vertical anti-symmetric mode 0.4235 0.445 Ϫ 4.8 First torsional symmetric mode 0.4989 0.525 Ϫ 5.0 First torsional anti-symmetric mode 0.6103 0.665 Ϫ 8.2

in terms of the warping stiffness (constant) Jw of the mode and the first anti-symmetric torsional vibration original bridge deck divided by (2b2), where b is the mode, respectively, of the Nanpu Bridge for different distance between the side girder and the centroid [see values of h in the triple-girder model. They are also Eq. (9)]. The conventional way of determining the warp- shown in Fig. 6. The warping effects are reflected in s a ing constant is to find the shear center of the deck cross- terms of the factors E1(h) and E1(h), which are section first, then calculate the principal sectorial area at defined as: each point of the cross-section, and finally integrate the f s (h) − f s (0) f a (h) − f a (0) sectorial area over the whole cross-section to obtain the s ϭ t1 t1 a ϭ t1 t1 E1(h) s , E1(h) a . (19) warping constant. If the open cross-section of the bridge f t1(0) f t1(0) deck is rather complicated, determination of its warping constant will be quite time-consuming. Therefore, to The torsional frequency estimation errors due to the s a facilitate the application of the proposed triple-girder ratio h are expressed by the factors G1(h) and G1(h), model, it is necessary to find a simple way for determin- which are defined as: ing the second moment of area about the z-axis J other z2 f s (h) − f s (100%) than through the warping constant. s ϭ t1 t1 a G1(h) s , G1(h) (20) Note that the two side girders in the triple-girder f t1(100%) model are usually set at the positions of two side girders f a (h) − f a (100%) of the original bridge deck. It is thus interesting to know ϭ t1 t1 . f a (100%) if the second moment of the original side girder includ- t1 ing a certain surrounding area can be used as the second moment of area of the side girder in the triple-girder Obviously, G1(100%) equals zero, indicating that the model. In this connection, this section first investigates warping torsional stiffness of the bridge deck is accu- the effects of the value of Jz2 in the triple-girder model rately included. G1(h) less than zero means that the on the torsional frequencies of the Nanpu Bridge. Based warping stiffness is underestimated whilst G1(h) larger on the results obtained, it is then checked whether the than zero indicates that the warping stiffness is overesti- aforementioned idea can be used to determine the value mated. of Jz2. It can be seen from Table 3 and Fig. 6 that, with the increasing value of h, the two torsional frequencies are 4.1. Parametric study increased. The increasing rate is large when h is small but becomes small when the value of h reaches more w w = 2 Denote h as Jz2/Jz2, where Jz2 Jw/(2b ) is the nor- than 80%. For an h value of 80%, the warping effects w s a malized warping constant, and calculate Jz2 of the bridge on f t1 and f t1 are 17.2% and 34.3%, respectively. These deck. Then, the change of value h gives a series of the values only underestimate the real warping effects by second moment of area of the side girder in the triple- 0.9% and 2.2%, respectively. On the other hand, the girder model. Clearly, if h is zero, the second moment possible maximum value of h is 122.4% when Jz2 takes of the side girder Jz2 is zero, which means no warping its possible maximum value equal to Jz/2. For this upper- s a stiffness taken into consideration in the computation of bound value, the warping effects on f t1 and f t1 are 19.1% the torsional frequencies. If h is one, the warping stiff- and 40.2%, respectively. These values only overestimate ness is fully taken into account as in the case study of the real warping effects by 0.7% and 2.1%, respectively. the Nanpu Bridge. Therefore, the variation of torsional Thus one may conclude that, for the Nanpu Bridge, if frequencies of the bridge with the ratio h reflects the the real warping torsional stiffness of the bridge deck is effects of the value Jz2 in the triple-girder model on the underestimated or overestimated by 20%, the first and torsional frequencies of the bridge. second torsional frequencies are underestimated or over- s Table 3 displays the two torsional frequencies, f t1 and estimated by only 0.9% and 2.2%, respectively. This a f t1, associated with the first symmetric torsional vibration indicates that the torsional frequencies are not very 中国科技论文在线 http://www.paper.edu.cn

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Table 3 Variations of warping effect and torsional frequency estimation error with ratio h

s s s a a a h (%) f t1 (Hz) E1(h) (%) G1 (%) f t1 (Hz) E1(h) (%) G1 (%)

0 0.4218 0.0 Ϫ 15.5 0.4443 0.0 Ϫ 27.2 1 0.4343 3.0 Ϫ 12.9 0.4596 3.4 Ϫ 24.7 3 0.4422 4.8 Ϫ 11.4 0.4737 6.6 Ϫ 22.4 5 0.4470 6.0 Ϫ 10.4 0.4831 8.7 Ϫ 20.8 10 0.4553 7.9 Ϫ 8.7 0.4992 12.4 Ϫ 18.2 15 0.4613 9.4 Ϫ 7.5 0.5157 16.1 Ϫ 15.5 20 0.4662 10.5 Ϫ 6.6 0.5258 18.3 Ϫ 13.8 30 0.4740 12.4 Ϫ 5.0 0.5433 22.3 Ϫ 11.0 40 0.4799 13.8 Ϫ 3.8 0.5575 25.5 Ϫ 8.7 50 0.4847 14.9 Ϫ 2.8 0.5694 28.2 Ϫ 6.7 60 0.4885 15.8 Ϫ 2.1 0.5797 30.5 Ϫ 5.0 70 0.4918 16.6 Ϫ 1.4 0.5887 32.5 Ϫ 3.5 80 0.4945 17.2 Ϫ 0.9 0.5966 34.3 Ϫ 2.2 90 0.4968 17.8 Ϫ 0.4 0.6052 36.2 Ϫ 0.8 100 0.4989 18.3 0.0 0.6103 37.4 0.0 110 0.5006 18.7 0.3 0.6162 38.7 1.0 122.4 0.5024 19.1 0.7 0.6229 40.2 2.1

Yangtze Bridge in Chongqing, and the Jingsha Yangtze Bridge in Hubei. Figs. 7(a) to (e) show the cross-sections of the five bridge decks. They are all open cross-sections but with different types of girders and slab edges. Com- posite decks are used in the Nanpu Bridge, the Yangpu Bridge and the Xupu bridge, whilst pre-stressed concrete decks are used in the Chongqing Second Yangtze Bridge and the Jingsha Yangtze Bridge. The Nanpu Bridge, the Chongqing Second Yangtze Bridge and the Jingsha Yangtze Bridge use H-type towers and two parallel cable planes to support the bridge deck. However, the Yangpu Bridge has upside-down Y-shaped towers and the Xupu Bridge has A-shaped towers, and thus there are two inclined cable planes for each bridge. The Yangpu Fig. 6. Variation of torsional frequencies with ratio h for the Bridge is located in downtown Shanghai over the Nanpu Bridge. Huangpu River. The full length of the bridge is 1088 m while the main span of the bridge is 602 m. The main sensitive to the value of Jz2 when this value is greater span of the Xupu Bridge is 590 m and its two side spans than a certain level. Thus, one may use the second are 202 m each. The main span of the Chongqing Second moment of the original side girder with a certain sur- Yangtze Bridge is 444 m and its two side spans are rounding area as the equivalent second moment of area 169 m each. The Jingsha Yangtze Bridge has a main of the side girder in the triple-girder model, so that the span of 500 m and two side spans of 200 m each. time-consuming calculation of the warping constant of For the five typical bridge decks shown in Fig. 7, each the bridge deck can be avoided. cross-section has a vertical symmetric axis. The horizontal z-axis and vertical y-axis consist of a pair of principal 4.2. Verification of simplified calculation centroid axes for the entire cross-section. The characteristic dimensions for girders and slabs of each cross-sec- To verify the idea mentioned above of using the tion, indicated in Fig. 7, are given in Table 4. On the second moment of the original side girder with certain basis of these dimensions, the normalized warping con- w = 2 surrounding area as the equivalent second moment of stant Jz2 Jw/(2b ) is calculated for each cross-section area of the side girder Jz2 in the triple-girder model, the with respect to its shear center. On the other hand, by five typical cable-stayed bridges in China are selected. trial and error, it is decided that the value of Jz2 is calcu- They are the Nanpu Bridge in Shanghai, as discussed in lated based on the area enclosed by the dashed box previous sections, the Yangpu Bridge in Shanghai, the shown in Fig. 7 about the z-axis. The area enclosed by Xupu Bridge in Shanghai, the Chongqing Second the dashed box is almost symmetric with respect to the 中国科技论文在线 http://www.paper.edu.cn

L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 1321

were then carried out, by using the single-girder model without warping effects, the triple-girder model with w Jz2, and the triple-girder model using Jz2. Tables 6 and 7 list, respectively, the two torsional frequencies with one corresponding to the first symmetric torsional vibration mode and the other to the first anti-symmetric torsional vibration mode. Clearly, these results confirm

that the second moment of area Jz2 can be used to replace w the normalized warping constant Jz2 in the triple-girder model for the cable-stayed bridges of typical open-section deck. The maximum deviation of the torsional frequency is less than 1% only. The results also show that, except for the Nanpu Bridge, the warping effects of the other four bridges are quite weak. This is because solid rectangular side girders are used in the Chongqing Second Yangtze Bridge and the Jingsha Yangtze Bridge, and inclined cable planes together with boxed side girders are used in the Yangpu Bridge and the Xupu Bridge. To gain more understanding of the relationships between the warping effect and bridge structural systems, the open section with two I-side girders of the Nanpu bridge deck was changed to the open section but with two boxed side girders like the Yangpu bridge deck. The modal analysis then shows that the warping effects are reduced from 18.3% to 7.3% for the first torsional frequency and from 37.4% to 19.6% for the second torsional frequency. Furthermore, if the open section with two boxed side girders of the Yangpu bridge deck is changed to the open section with two I-side girders like the Nanpu bridge deck, the modal analysis shows that the warping effects are increased from 1.4% to 8.1% for the first torsional frequency and from 2.8% to 8.6% for the second torsional frequency. Clearly, warping effects depend on bridge cable systems and bridge girder con- figurations.

5. Conclusions

A triple-girder model has been proposed in this paper Fig. 7. Open cross-sections of five typical cable-stayed bridge decks: (a) Nanpu Bridge; (b) Yangpu Bridge; (c) Xupu Bridge; (d) Chongqing for modal analysis of cable-stayed bridges including Second Yangtze Bridge; (e) Jingsha Yangtze Bridge. warping effects. The model has been verified to some extent through the case study of the Nanpu cable-stayed bridge in China, in which the natural frequencies com- vertical axis of the side girder. The calculated results of puted using the triple-girder model were found to be in w Jz2 and Jz2 are listed in Table 5. It is seen that the differ- better agreement with the measured results than results w ence between the values of Jz2 and Jz2 ranges from 0.2% from the conventional single-girder model without warp- to 15.5% for the five bridge decks concerned. According ing effects. to the results from the last section — that the torsional There are two main advantages in using the proposed frequencies are not very sensitive to the value of Jz2 triple-girder model for modal analysis of cable-stayed when this value is greater than 80% of the normalized bridges. First, the model makes it possible to include warping constant, it is thus expected that the use of Jz2 warping effects of the bridge deck by using conventional w defined in this study to replace Jz2 in the triple-girder beam elements of 12 degrees of freedom only, so that model may not significantly affect the estimation of the commercial finite element programs familiar to engin- torsional frequencies. eers can still be used and the user-friendly features in Modal analyses of five typical cable-stayed bridges the single-girder beam element model remain. Second, 中国科技论文在线 http://www.paper.edu.cn

1322 L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323

Table 4 Characteristic dimensions of bridge deck cross-sections

Dimension Nanpu Yangpu Xupu Chongqing Jingsha

b (m) 12.500 12.500 16.625 9.050 12.650

b1 (m) 2.955 2.675 – 2.000 –

t1 (m) 0.260 0.260 0.260 0.280 0.320

b2 (m) 0.800 2.300 2.700 – –

t2 (m) 0.035 0.025 0.025 – –

b3 (m) 0.800 2.300 2.700 – –

t3 (m) 0.060 0.035 0.035 – –

h0 (m) 2.100 2.700 2.700 2.500 2.000

t4 (m) 0.02 0.016 0.02 – –

b0 (m) – 1.500 1.900 1.700 1.900

Table 5 w Comparison of section properties Jz2 and Jz2

Section property Nanpu Yangpu Xupu Chongqing Jingsha

w 4 Jz2 (m ) 0.2728 0.6594 0.7541 3.6862 2.2207 4 Jz2 (m ) 0.2963 0.7226 0.8710 3.6675 2.2161 = w h Jz2/Jz2 (%) 108.6 110.0 115.5 99.5 99.8

Table 6 Natural frequencies of ﬁve bridges for their ﬁrst symmetrical torsional modes

Bridge (1) Nanpu Yangpu Xupu Chongqing Jingsha

Single-girder model (2) 0.4218 0.5096 0.6081 0.4618 0.3803 Triple-girder model with w (3) 0.4989 0.5169 0.6144 0.4690 0.3848 Jz2 Triple-girder model with (4) 0.5004 0.5175 0.6150 0.4690 0.3848 Jz2 [(3) Ϫ (2)]/(2) (%) (5) 18.3 1.4 1.0 1.6 1.2 [(4) Ϫ (2)]/(2) (%) (6) 18.6 1.6 1.1 1.6 1.2 [(4) Ϫ (3)]/(3) (%) (7) 0.3 0.1 0.1 0.0 0.0

Table 7 Natural frequencies of ﬁve bridges for their ﬁrst anti-symmetric torsional modes

Bridge (1) Nanpu Yangpu Xupu Chongqing Jingsha

Single-girder model (2) 0.4443 0.6040 0.6769 0.6317 0.5073 Triple-girder model with w (3) 0.6103 0.6210 0.6940 0.6545 0.5243 Jz2 Triple-girder model with (4) 0.6154 0.6228 0.6956 0.6544 0.5243 Jz2 [(3) Ϫ (2)]/(2) (%) (5) 37.4 2.8 2.5 3.6 3.4 [(4) Ϫ (2)]/(2) (%) (6) 38.5 3.1 2.8 3.6 3.4 [(4) Ϫ (3)]/(3) (%) (7) 0.8 0.3 0.2 0.0 0.0

the use of three girders makes it easy to assign the longi- present is basically a linear model for modal analysis of tudinal, lateral, vertical and pure torsional and warping the bridge. Thus, further study is required on how to stiffness properties of the original bridge deck to the new apply the triple-girder model to wind-induced lateral– model. However, the proposed triple-girder model at torsional buckling and ﬂutter problems of long-span 中国科技论文在线 http://www.paper.edu.cn

L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 1323 cable-stayed bridges, in which the large twist defor- al. of the Department of Bridge Engineering of Tongji mation of the bridge deck and geometric non-linear University. The former provided part of the structural analysis should be considered. data on the Chongqing Second Yangtze Bridge and the To further facilitate the application of the triple-girder Jingsha Yangtze Bridge, whilst the others provided the model, this paper has proposed use of the second results of the ambient vibration measurement of Nanpu moment of the original side girder with a certain sur- Bridge. Support from the Hong Kong Polytechnic Uni- rounding area as the equivalent second moment of area versity through a scholarship for the first author to con- of the side girder in the triple-girder model, to avoid the tinue his study is also appreciated. time-consuming calculation of the warping constant. This idea was based on results from a parametric study of the Nanpu Bridge, which demonstrated that a less than References 20% deviation from the real warping stiffness of a bridge deck affects the torsional frequencies only slightly. This [1] Ereiba HA. The dynamic behavior of cable-stayed bridges. PhD idea was also verified to some extent through the modal thesis. Sheffield: Department of Civil and Structural Engineering, University of Sheffield, 1979. analyses of five different types of cable-stayed bridge [2] Yiu PKA, Brotton DM. Mathematical modeling of cable-stayed in China. bridges for computer analysis. In: Proc. Int. Conf. on Cable- The modal analysis of the five cable-stayed bridges in Stayed Bridge, Bangkok, Thailand, 1987:261–75. China also highlighted that the extent of warping effects [3] Kanok-Nukulchai W, Yiu PKA, Brotton DM. Mathematical depends greatly on the type of bridge deck and the type modeling of cable-stayed bridges. Struct Eng Int 1992;2:108–13. [4] Branco EA, Azevedo J, Ritto-Corretia M, Campos-Costa A. of bridge tower–cable system. For a bridge of H-type Dynamic analysis of the International Guadiana Bridge. Struct towers with two parallel cable planes and a bridge deck Eng Int 1993;4:240–4. of weak pure torsional stiffness, such as the Nanpu [5] Yang YB, McGuire W. A stiffness matrix for geometric non- Bridge, inclusion of the warping stiffness (constant) in linear analysis. J Struct Eng, ASCE 1986;112(ST4):853–77. the modal analysis is necessary. For a bridge of H-type [6] Yang YB, McGuire W. Joint rotations and geometric non-linear analysis. J Struct Eng, ASCE 1986;112(ST4):879–905. towers with two parallel cable planes but with solid side [7] Boonyapinyo V, Yamada H, Miyata T. Wind-induced nonlinear girders, such as the Chongqing Second Yangtze Bridge lateral–torsional buckling of cable-stayed bridges. J Struct Eng, and the Jingsha Yangtze Bridge, the warping stiffness ASCE 1994;120(2):486–506. effects on the modal properties are relatively small. For [8] Wilson JC, Gravelle W. Modeling of a cable-stayed bridge for a bridge with upside-down Y-shaped towers or A-shaped dynamic analysis. Earthquake Eng Struct Dyn 1991;20:707–21. [9] Craig RR. Mechanics of materials. New York, NY: John Wiley & towers with two inclined cable planes and a bridge deck Sons, Inc, 1996. of two side box girders, such as the Yangpu Bridge and [10] Shanghai Nanpu Bridge vibration measurement report. Shanghai, the Xupu Bridge, the warping stiffness effects on the People’s Republic of China: Department of Bridge Engineering, modal properties are also relatively small. Tongji University, 1992.

Acknowledgements

The authors would like to express their sincere thanks to Ms H. Ha, Professor J.J. Shi and Mr G.Y. Zhang et